Smooth quotients of abelian surfaces by finite groups that fix the origin

Abstract

Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$ is smooth. In particular, we prove that $A=E^2$ with $E$ an elliptic curve and that $A/G\simeq {\mathbb{P}}^2$ in all cases. Moreover, for fixed $E$, there are only finitely many pairs $(E^2,G)$ up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors.

arXiv comment14 pages. arXiv admin note: text overlap with arXiv:1801.00028

References in corpus (2)

• Discrete complex reflection groups
• Smooth quotients of abelian varieties by finite groups